Find the center, foci, and vertices of each ellipse. Graph each equation. 4 x^2+3 y^2+8 x-6 y=5

Name: Problem 50, Chapter 11: Algebra and Trigonometry
Uploaded: 2021-08-17T19:38:53-08:00
Duration: 5 min 12 s
Channel: David Baker
Description: Find the center, foci, and vertices of each ellipse. Graph each equation. 4 x^2+3 y^2+8 x-6 y=5

Find the center, foci, and vertices of each ellipse. Graph...

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse’s equation exposes the key features of the ellipse, such as its center, lengths of the semi-major and semi-minor axes, and orientation. This form is instrumental in analyzing and graphing the ellipse easily.

Ellipse

An ellipse is a type of conic section defined as the set of all points for which the sum of the distances to two fixed points (foci) is constant. It is characterized by its oval shape and symmetry about both its major and minor axes.

Completing the Square

Completing the square is an algebraic method used to transform quadratic expressions into a perfect square plus or minus a constant. This technique is crucial for rewriting the general equation of conic sections into their standard forms, thereby revealing important features like the center of an ellipse.

Center, Vertices, and Foci

The center of an ellipse is the midpoint of both the major and minor axes, while the vertices are the points where the ellipse intersects the major axis, representing the maximum distances from the center. The foci are two interior points along the major axis, whose positions are determined by the ellipse's eccentricity, and they play a central role in the geometric definition of the ellipse.

Transcript

00:01 In this ellipse, we're going to complete the square to find the center, vertices, and fosci.

00:05 We'll start by grouping the x terms together and the y terms together.

00:14 And now we'll factor out the leading coefficient of the x squared.

00:20 Leave room to add something to complete the square.

00:23 Factor out the three.

00:28 And now we can add the magic number to complete the square.

00:32 We find the magic number by taking half of the linear coefficient and squaring it.

00:37 So in this case, that's going to be 2 over 2 squared.

00:41 The magic number is a 1.

00:43 So i'll add 1 here.

00:45 And here, it's negative 2 over 2 square that.

00:50 It's 1 again.

00:52 So in both parts, i've added 1.

00:55 Now i need to do that to the left, but i need to be careful here because i've already factored out of 4.

00:59 I'm not really adding 1.

01:01 I'm adding 4 times 1, which is 4.

01:04 And here i've added a 1, but i've really added 3 times 1, which is 3.

01:10 So now the equation is balanced.

01:12 I've done something useful on the left, and i've changed the right just to maintain equality.

01:18 Now i can complete the square, and i get 4 times x plus 1 squared, plus 3 times y and minus 1 squared, equals the numbers on the right looks like they sum to 5 plus 4 is 9 plus 3 is 12.

01:35 To make this look like an ellipse in standard form, i need the right side to equal 1.

01:39 So i'm going to divide everything by 12.

01:44 So i'll divide both sides by 12, which gives me a right -hand side that equals 1.

01:49 So in this ellipse, the center is located at negative 1, comma, 1.

01:54 I'm taking the opposite of the numbers i see in the numerator.

01:58 A squared is equal to 4, b squared is equal to 3.

02:04 So a squared is the bigger number.

02:06 We find a squared located under y.

02:10 B squared is the smaller denominator.

02:12 We find b squared located under x.

02:14 That means this ellipse has a major axis that is vertical because the bigger number was under y.

02:24 Finally, we can find c, which will help us with the fosite.